In English your sentence would be "No one can work here," but I don't think that's what it says in Czech here. I wouldn't normally say "not anyone." We might, however, say "Not just anyone can work here." This means that there are some people more suited to the job than others.
"Not anyone can work here" has a number of reports in the system. I agree that it does not work, and that "Not JUST anyone can work here" would be a very natural translation on the English side. I will add it, IF the Czech natives on the team feel it is not too far from the original sentence.
He must mean komkoli from the previous exercise. It is the locative of kdokoli so it went after o: Je velmi špatné o komkoli tohle říkat. "It is very bad to say this about anyone/who(m)(so)ever." Kdokoli is the nominative so it must be the subject only, just like kdo for 'who' as you say.